// Numbas version: finer_feedback_settings
{"name": "Leicester: Differentiation 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Leicester: Differentiation 4", "tags": ["chain rule", "differentiation", "quotient rule"], "advice": "\n  <p>c) Let $\\displaystyle \\simplify[std]{u={u}*e^({c}x)+{1-u}*ln({c}*x)}$</p>\n  <p>So we have: $\\displaystyle \\frac{du}{dx}=\\simplify[std]{{u*c}*e^({c}x)+{1-u}/x}$ and&nbsp;$\\displaystyle f(u)=\\simplify[std]{{t[0]}*sin(u)+{t[1]}*cos(u)}\\Rightarrow \\frac{df}{du}=&nbsp;\\simplify[std]{{t[0]}*cos(u)-{t[1]}*sin(u)}$.</p>\n  <p>By the Chain Rule \\[\\begin{eqnarray*} \\frac{df}{dx}&amp;=&amp;\\frac{df}{du}\\times\\frac{du}{dx}\\\\&amp;=&amp;\\simplify[std]{{t[0]}*cos(u)-{t[1]}*sin(u)}\\times\\simplify[std]{{u*c}*e^({c}x)+{1-u}/x}\\\\&amp;=&amp;\\simplify[std]{{t[0]}*({u}*{c}e^({c}x)+{1-u}/x)*cos({u}*e^({c}x)+{1-u}*ln({c}*x))-{t[1]}*({u}*{c}e^({c}x)+{1-u}/x)*sin({u}*e^({c}x)+{1-u}*ln({c}*x))}\\end{eqnarray*}\\]</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"prompt": "\n      <p>$\\displaystyle f(x)=\\simplify[all]{(x^{a}-{b})/(x^{-c}+{d})}$</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers, not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.5, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all", "marks": 1.0, "answer": "({a+c}x^{a-c-1}+{a*d}x^{a-1}-{b*c}x^{-c-1})/(x^{-c}+{d})^2", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>$\\displaystyle f(x)=\\simplify[std]{(e^({a}x)-e^({b}x))/( e^({c}x)+e^({d}x))}$</p>\n      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers, not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [1.0, 2.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std,!noLeadingMinus", "marks": 1.0, "answer": "({a-c}e^({a+c}x)+{a-d}e^({a+d}x)+{c-b}e^({b+c}x)+{d-b}e^({b+d}x))/(e^({c}x)+e^({d}x))^2", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>$f(x)=\\simplify[std]{{t[0]}*sin({u}*e^({c}x)+{1-u}*ln({c}*x))+{t[1]}*cos({u}*e^({c}x)+{1-u}*ln({c}*x))}$.</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.5, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{t[0]}*({u}*{c}e^({c}x)+{1-u}/x)*cos({u}*e^({c}x)+{1-u}*ln({c}*x))-{t[1]}*({u}*{c}e^({c}x)+{1-u}/x)*sin({u}*e^({c}x)+{1-u}*ln({c}*x))", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "\n    <p>Differentiate the following functions $f(x)$ with respect to $x$.</p>\n    <p>Input all numbers as fractions or integers, not as decimals.</p>\n    \n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-5..5 except 0)", "name": "a"}, "c": {"definition": "random(1..7 except d)", "name": "c"}, "b": {"definition": "random(-5..5 except [a,0])", "name": "b"}, "d": {"definition": "random(2..9 except 0)", "name": "d"}, "f": {"definition": "random(2..9)", "name": "f"}, "ee": {"definition": "random(0..10)", "name": "ee"}, "u": {"definition": "random(0,1)", "name": "u"}, "t": {"definition": "switch(v=0,[1,0],[0,1])", "name": "t"}, "v": {"definition": "random(0,1)", "name": "v"}}, "metadata": {"notes": "", "description": "<p>Examples on differentiation using the quotient rule and chain rule.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "resources": []}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}